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A060255 Smaller of twin primes {p, p+2} whose average p+1 = k*q is the least multiple of the n-th primorial number q such that k*q-1 and k*q+1 are twin primes.

%I A060255 #26 Apr 19 2025 15:42:21
%S A060255 3,5,29,419,2309,180179,4084079,106696589,892371479,103515091679,
%T A060255 4412330782859,29682952539239,22514519501013539,313986271960080719,
%U A060255 22750921955774182169,912496437361321252439,26918644902158976946979,1290172194953476680815969,1901713815361424627522739779
%N A060255 Smaller of twin primes {p, p+2} whose average p+1 = k*q is the least multiple of the n-th primorial number q such that k*q-1 and k*q+1 are twin primes.
%C A060255 a(349) has 1001 digits. - _Michael S. Branicky_, Apr 19 2025
%H A060255 Michael S. Branicky, <a href="/A060255/b060255.txt">Table of n, a(n) for n = 1..348</a>
%F A060255 a(n) = p = k(n)*q(n)-1, where q(n)=A002110(n) and k(n)=A060256(n) is the smallest integer whose multiplication by the n-th primorial yields p+1.
%e A060255 a(13) = -1 + (2*3*5*7*...*41)*k(13) = 304250263527210*74 and {22514519501013539, 22514519501013542} are the corresponding primes; k(13)=74 is the smallest suitable multiplier. Twin primes obtained from primorial numbers with k=1 multiplier seem to be much rarer (see A057706).
%e A060255 For j=1,2,3,4,5,6, a(j)=A001359(1), A059960(1), A060229(1), A060230(1), A060231(1), A060232(1) respectively.
%o A060255 (PARI) a(n) = {my(q = prod(k=1, n, prime(k))); for(k=1, oo, if (isprime(q*k-1) && isprime(q*k+1), return(q*k-1)););} \\ _Michel Marcus_, Jul 10 2018
%o A060255 (Python)
%o A060255 from itertools import count
%o A060255 from sympy import primorial, isprime
%o A060255 def a(n):
%o A060255     p = primorial(n)
%o A060255     return next(m-1 for m in count(p, p) if isprime(m-1) and isprime(m+1))
%o A060255 print([a(n) for n in range(1, 20)]) # _Michael S. Branicky_, Apr 18 2025
%Y A060255 Cf. A001359, A002110, A006794, A014545, A057706, A059960, A060229, A060230, A060231, A060232.
%K A060255 nonn
%O A060255 1,1
%A A060255 _Labos Elemer_, Mar 22 2001
%E A060255 a(2)=5 corrected by _Ray Chandler_, Apr 03 2009
%E A060255 a(18) and beyond from _Michael S. Branicky_, Apr 18 2025